Porous sets that are Haar null, and nowhere approximately differentiable functions
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- by Jan Kolář PDF
- Proc. Amer. Math. Soc. 129 (2001), 1403-1408 Request permission
Abstract:
We define a new notion of “HP-small” set $A$ which implies that $A$ is both $\sigma$-porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on $[0,1]$ which have finite unilateral approximate derivative at a point $x\in [0,1]$ is HP-small, as well as its projections onto hyperplanes. As a corollary, the same is true for the set of all Besicovitch functions. Also, the set of continuous functions on $[0,1]$ which are Hölder at a point is HP-small.References
- Valeriu Anisiu, Porosity and continuous, nowhere differentiable functions, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 1, 5–14 (English, with English and French summaries). MR 1230703
- S. Banach, Über die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174–179.
- Andrew Bruckner, Differentiation of real functions, 2nd ed., CRM Monograph Series, vol. 5, American Mathematical Society, Providence, RI, 1994. MR 1274044, DOI 10.1090/crmm/005
- J. P. R. Christensen, Topology and Borel structure, North-Holland Mathematics Studies, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory. MR 0348724
- P. M. Gandini and A. Zucco, Porosity and typical properties of real-valued continuous functions, Abh. Math. Sem. Univ. Hamburg 59 (1989), 15–21. MR 1049878, DOI 10.1007/BF02942311
- Brian R. Hunt, The prevalence of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 122 (1994), no. 3, 711–717. MR 1260170, DOI 10.1090/S0002-9939-1994-1260170-X
- Eva Matoušková, The Banach-Saks property and Haar null sets, Comment. Math. Univ. Carolin. 39 (1998), no. 1, 71–80. MR 1622974
- S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92–94.
- D. L. Renfro, Some supertypical nowhere differentiability results for ${\mathcal C}[0,1]$, Doctoral Dissertation, North Carolina State University, 1993.
- L. Zajíček, Porosity and $\sigma$-porosity, Real Anal. Exchange 13 (1987/88), no. 2, 314–350. MR 943561
Additional Information
- Jan Kolář
- Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
- Email: kolar@karlin.mff.cuni.cz
- Received by editor(s): August 9, 1999
- Published electronically: October 25, 2000
- Additional Notes: The author was supported by the grants GAUK 165/99 and CEZ:J13/98:113200007.
- Communicated by: David Preiss
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1403-1408
- MSC (1991): Primary 26A27, 28C20, 26A16, 26A24
- DOI: https://doi.org/10.1090/S0002-9939-00-05811-1
- MathSciNet review: 1814166